In addition to talking up the company's ever-imminent merger with American Airlines, Parker, along with President Scott Kirby, addressed one of consumers' biggest gripes: Baggage fees.

Though they danced around a question on whether carry-on baggage fees were on the horizon (Kirby: "I think we will continue to experiment."), Parker did acknowledge the company's role in perpetuating the cluttered cabins and delayed boarding times that have passengers riled up:

"There are indeed bags that are getting down to the aircraft that should have been checked. We acknowledge that we've created the problem. We're not trying to blame the customer," he said. "We created this problem by charging for checking and not charging if you bring it on. People will do things to avoid paying that fee."

Kirby, on the other hand, said there was no simple solution:

"We acknowledge and recognize that there are more (carry-on) bags on the airplanes than there have been historically. As an airline, we've invested in an operations crew to run the operation better but it still creates that anxiety. I'm not sure what the right answer is. The right answer also depends on what the competitive environment is."

While wildly unpopular with passengers, some airlines like Spirit have had marked success with charging for carry-on bags. Spirit charges $30 for carry-on bags vs. $28 for the first checked bag, which helps deter people from cramming bags on the flight to save cash (those fees are set to rise to $35 and $30 in November).